Planetary radii are derived for 218 exoplanets orbiting 161 M dwarf stars. Stellar radii are based on an analysis of APOGEE high-resolution near-IR spectra for a subsample of the M dwarfs; these results are used to define a stellar radius-M_Ks_ calibration that is applied to the sample of M-dwarf planet hosts. The planetary radius distribution displays a gap over Rp~1.6-2.0R_{Earth}_, bordered by two peaks at Rp~1.2-1.6R_{Earth}_ (super-Earths) and 2.0-2.4R_{Earth}_ (sub-Neptunes). The radius gap is nearly constant with exoplanetary orbital period (a power-law slope of m=+0.01_-0.04_^+0.03^), which is different (2{sigma}-3{sigma}) from m~-0.10 found previously for FGK dwarfs. This flat slope agrees with pebble accretion models, which include photoevaporation and inward orbital migration. The radius gap as a function of insolation is approximately constant over the range of Sp~20-250S_{Earth}_. The Rp-Porb plane exhibits a sub-Neptune desert for Porb<2-days, which appears at Sp>120S_{Earth}_, being significantly smaller than Sp>650S_{Earth}_ found in the FGK planet-hosts, indicating that the appearance of the sub-Neptune desert is a function of host- star mass. Published masses for 51 exoplanets are combined with our radii to determine densities, which exhibit a gap at {rho}_p_~0.9{rho}_{Earth}_, separating rocky exoplanets from sub-Neptunes. The density distribution within the sub-Neptune family itself reveals two peaks, at {rho}_p_~0.4{rho}_{Earth}_ and ~0.7{rho}_{Earth}_. Comparisons to planetary models find that the low-density group are gas-rich sub-Neptunes, while the group at <{rho}_p_>~0.7{rho}_{Earth}_ likely consists of volatile-rich water worlds.